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A note on a characterization of generalized quaternion 2-groups. (Caractérisation des 2-groupes de quaternions généralisés.) (English. French summary) Zbl 1303.20019
Summary: In this note, we answer an open problem posed by M. Tărnăceanu [in C. R., Math., Acad. Sci. Paris 348, No. 13-14, 731-733 (2010; Zbl 1205.20024)], and obtain that the generalized quaternion 2-groups are the unique finite noncyclic groups whose posets of conjugacy classes of cyclic subgroups have breaking points.

20D30 Series and lattices of subgroups
20D15 Finite nilpotent groups, \(p\)-groups
Zbl 1205.20024
Full Text: DOI
[1] Călugăreanu, G. G.; Deaconescu, M., Breaking points in subgroup lattices, (Campbell, C. M.; Robertson, E. F.; Smith, G. C., Proceedings of Groups St. Andrews 2001 in Oxford, vol. 1, (2003), Cambridge University Press Cambridge, UK), 59-62 · Zbl 1062.20028
[2] Fein, B.; Kantor, W. M.; Schacher, M., Relative Brauer groups. II, J. Reine Angew. Math., 328, 39-57, (1980) · Zbl 0457.13004
[3] Huppert, B., Endliche gruppen, I, (1967), Springer-Verlag Berlin, Heidelberg, New York · Zbl 0217.07201
[4] Tărnăuceanu, M., Groups determined by posets of subgroups, (2006), Ed. Matrix Rom Bucuresti, Romania · Zbl 1123.20001
[5] Tărnăuceanu, M., A characterization of generalized quaternion 2-groups, C. R. Acad. Sci. Paris, Ser. I, 348, 731-733, (2010) · Zbl 1205.20024
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